Definition:Finer Filter on Set

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Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal F, \mathcal F' \subset \mathcal P \left({S}\right)$ be two filters on $S$.

Let $\mathcal F \subseteq \mathcal F'$.

Then $\mathcal F'$ is finer than $\mathcal F$.

Strictly Finer

Let $\mathcal F \subset \mathcal F'$, that is, $\mathcal F \subseteq \mathcal F'$ but $\mathcal F \ne \mathcal F'$.

Then $\mathcal F'$ is strictly finer than $\mathcal F$.

Also known as

A finer filter than $\mathcal F$ can also be referred to as a superfilter of $\mathcal F$.

Also see